The vector a= i -2j and b= -3i + j. We have to determine `beta` such that a + `beta`b is parallel to -i - 3j and `mu` if `mu`*a + b is parallel to 2i + j.

This can be done using the property that the cross product of two parallel vectors is...

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The vector a= i -2j and b= -3i + j. We have to determine `beta` such that a + `beta`b is parallel to -i - 3j and `mu` if `mu`*a + b is parallel to 2i + j.

This can be done using the property that the cross product of two parallel vectors is equal to 0.

For vectors A = a1*i + a2*j and B = b1*i + b2*j, A x B = (a1*b2 - a2*b1)k

a + `beta`*b = i - 2j + `beta`*[-3i + j] = i - 3*`beta`*i -2j + `beta`*j = i(1 - 3`beta`) + j(`beta` -2)

Taking the cross product of this with -i - 3j we get (1 - 3`beta`)(-3) - (`beta` - 2)(-1) = 0

=> -3 + 9`beta` + `beta` - 2 = 0

=> 10`beta` - 5 = 0

=> `beta` = 1/2

Also `beta` can also be -1/2

` ``mu` *a + b = `mu` *(i - 2j) + (-3i + j) = `mu`*i - `mu`*2j - 3i + j = i(`mu` - 3) + j(1 - 2*`mu`)

Taking the cross product of this with 2i + j we get (`mu` - 3)*1 - (1 - 2`mu`)*2 = 0

=> `mu` - 3 - 2 + 4`mu` = 0

=> 5`mu` - 5 = 0

=> `mu` = 1

Also `mu` can be equal to -1.

**The required values are `beta` = 1/2 and `beta` = -1/2 and `mu` = 1 and `mu` = -1**